A researcher claims that the distribution of adult heights in a population is normally distributed. To test this claim, a sample of $150$ adults is taken, and their heights (in inches) are grouped into the following class intervals:

Use $\alpha =0.05$ to test the claim that the heights are normally distributed if ${\chi }^2=3.72$.

A health researcher claims that the distribution of gym attendance across the week is as follows:

You randomly survey $500$ gym members and ask which day of the week they most often go to the gym. The results are:

At $\alpha =0.05$, test the researcher’s claim.

Fill in the blanks.

In order to perform a goodness-of-fit test, the two main requirements are that the data must be in the form of ____________________ and each expected frequency should be at least ___.

A hospital administrator claims that emergency room (ER) visits occur equally across all days of the week. To test this, a random sample of $840$ ER visits from the past month is analyzed. The number of visits per day is recorded below:

Test the administrator’s claim at the $\alpha =0.05$ significance level.

Find the critical values ${\chi }_{1-\frac{\alpha }{2}}^2$ and ${\chi }_{\frac{\alpha }{2}}^2$ for a confidence level of $95\%$ with sample size $n=15$.

A wildlife biologist recorded the number of sightings of a bird species at different elevation ranges. The data is:

Test the claim that the distribution of sightings is normally distributed using a chi-square goodness-of-fit test at the $0.05$ significance level.

A researcher claims that the distribution of people’s preferred grocery shopping methods has changed from a past survey. According to the past survey, the distribution was:

In-store shopping: $50\%$

Online pickup: $20\%$

Home delivery: $15\%$

Subscription service: $15\%$

To test this claim, a sample of $400$ people was surveyed with the following results:

Decide whether to reject or fail to reject the null hypothesis if $\alpha =0.05$.

Find the expected frequency for $n=150$ and $p_i=0.6$.

A researcher collected data on the drying times (in minutes) of four types of wall paint: matte, satin, semi-gloss, and gloss. The data were collected under controlled temperature and humidity conditions. The drying times (in minutes) are listed below:

Matte: $38,41,43,39,40$

Satin: $44,47,45,46,43$

Semi-gloss: $50,49,51,48,52$

Gloss: $42,40,41,43,44$

At the $0.05$ significance level, use the Kruskal-Wallis test to determine whether there is a significant difference in the median drying times among the four paint types.

Consider the contingency table below. Compute the marginal frequencies for each row and column, and determine the expected frequency for each cell, assuming the variables are independent.

State whether the following statement is true or false. In a chi-square test of independence, a large difference between observed and expected frequencies suggests that the variables are dependent.

A hospital records the last digits of $3,000$ newborns' birth weights in grams. The frequencies for digits $0$ through $9$ are shown below:

What is the chi-square test statistic to determine whether the last digits are uniformly distributed?

A hospital administrator wants to know if there is a relationship between patient satisfaction (Satisfied, Unsatisfied) and type of insurance (Private, Public, None). The observed frequencies are:

At $\alpha =0.05$, test the claim that patient satisfaction and type of insurance are dependent.

Researchers collected data on the number of male and female patients admitted to a hospital during $3$ different shifts (Morning, Afternoon, Night). At $\alpha =0.01$, can you conclude that gender and shift are related?

A researcher surveys a random sample of employees from two different companies regarding their satisfaction with workplace wellness programs. The responses are categorized as 'Satisfied', 'Neutral', or 'Dissatisfied'. The data are shown in the contingency table below:

At $\alpha =0.05$, can you conclude that satisfaction with wellness programs is related to the company of employment? Perform a chi-square test for independence.